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Stable numerical differentiation: when is it possible
 Jour. Korean SIAM
"... Abstract. Two principally different statements of the problem of stable numerical differentiation are considered. It is analyzed when it is possible in principle to get a stable approximation to the derivative f ′ given noisy data fδ. Computational aspects of the problem are discussed and illustrate ..."
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Cited by 11 (8 self)
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Abstract. Two principally different statements of the problem of stable numerical differentiation are considered. It is analyzed when it is possible in principle to get a stable approximation to the derivative f ′ given noisy data fδ. Computational aspects of the problem are discussed and illustrated by examples. These examples show the practical value of the new understanding of the problem of stable differentiation. 1.
Continuous modified Newton’stype method for nonlinear operator equations
 ANN. DI MAT. PURE APPL
, 2002
"... A nonlinear operator equation F (x) = 0, F: H → H, in a Hilbert space is considered. Continuous Newton’stype procedures based on a construction of a dynamical system with the trajectory starting at some initial point x0 and becoming asymptotically close to a solution of F (x) = 0 as t → + ∞ are d ..."
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Cited by 3 (2 self)
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A nonlinear operator equation F (x) = 0, F: H → H, in a Hilbert space is considered. Continuous Newton’stype procedures based on a construction of a dynamical system with the trajectory starting at some initial point x0 and becoming asymptotically close to a solution of F (x) = 0 as t → + ∞ are discussed. Wellposed and illposed problems are investigated.
Dynamical systems method for . . .
, 2004
"... Consider an operator equation F (u) = 0 in a real Hilbert space. The problem of solving this equation is illposed if the operator F ′ (u) is not boundedly invertible, and wellposed otherwise. A general method, dynamical systems method (DSM) for solving linear and nonlinear illposed problems in a ..."
Abstract

Cited by 3 (1 self)
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Consider an operator equation F (u) = 0 in a real Hilbert space. The problem of solving this equation is illposed if the operator F ′ (u) is not boundedly invertible, and wellposed otherwise. A general method, dynamical systems method (DSM) for solving linear and nonlinear illposed problems in a Hilbert space is presented. This method consists of the construction of a nonlinear dynamical system, that is, a Cauchy problem, which has the following properties: 1) it has a global solution, 2) this solution tends to a limit as time tends to infinity, 3) the limit solves the original linear or nonlinear problem. New convergence and discretization theorems are obtained. Examples of the applications of this approach are given. The method works for a wide range of wellposed problems as well.
A continuous Newtontype method for unconstrained optimization
"... In this paper, we propose a continuous Newtontype method in the form of an ordinary differential equation by combining the negative gradient and Newton’s direction. It is shown that for a general function f(x), our method converges globally to a connected subset of the stationary points of f(x) und ..."
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Cited by 1 (1 self)
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In this paper, we propose a continuous Newtontype method in the form of an ordinary differential equation by combining the negative gradient and Newton’s direction. It is shown that for a general function f(x), our method converges globally to a connected subset of the stationary points of f(x) under some mild conditions; and converges globally to a single stationary point for a real analytic function. The method reduces to the exact continuous Newton method if the Hessian matrix of f(x) is positive definite. The convergence of the new method on the set of standard test problems in the literature are also reported.
unknown title
"... Consider an operator equation F (u) = 0 in a real Hilbert space. The problem of solving this equation is illposed if the operator F ′ (u) is not boundedly invertible, and wellposed otherwise. A general method, dynamical systems method (DSM) for solving linear and nonlinear illposed problems in a ..."
Abstract
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Consider an operator equation F (u) = 0 in a real Hilbert space. The problem of solving this equation is illposed if the operator F ′ (u) is not boundedly invertible, and wellposed otherwise. A general method, dynamical systems method (DSM) for solving linear and nonlinear illposed problems in a Hilbert space is presented. This method consists of the construction of a nonlinear dynamical system, that is, a Cauchy problem, which has the following properties: 1) it has a global solution, 2) this solution tends to a limit as time tends to infinity, 3) the limit solves the original linear or nonlinear problem. New convergence and discretization theorems are obtained. Examples of the applications of this approach are given. The method works for a wide range of wellposed problems as well. 1